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\pub{2009}{1}{3}{1}
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\topic{Lecture 2.3 \\Matrix\\ \scriptsize Linear Equation System (05 Nov 2009)}
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\section{Non-homogeneous linear equation system}
Let the system of $m$ linear  equations in $n$ unknowns is:
\begin{equation} \label{nonhomoeq}
\begin{array}{cc}
a_{11}x_1 + a_{12}x_2 + \ldots + a_{1n}x_n &=b_1 \\
a_{21}x_1 + a_{22}x_2 + \ldots + a_{2n}x_n &=b_2 \\
\vdots &  \\
a_{m1}x_1 + a_{m2}x_2 + \ldots + a_{mn}x_n &=b_m \\
\end{array}
\end{equation}
In matrix form the system \ref{nonhomoeq} may be written as:
\[ AX = B\]
Then the matrix
\[
A = \left[%
\begin{array}{rrrr}
a_{11} & a_{12} & \ldots & a_{1n} \\
a_{21} & a_{22} & \ldots & a_{2n} \\
\vdots & \vdots & \vdots & \vdots \\
a_{m1} & a_{m2} & \ldots & a_{mn} \\
\end{array}%
\right]
\]
is called as \emph{coefficient matrix} and the matrix
\[
(A|B) = \left[%
\begin{array}{ccccc}
a_{11} & a_{12} & \ldots & a_{1n} & b_1\\
a_{21} & a_{22} & \ldots & a_{2n} & b_2\\
\vdots & \vdots & \vdots & \vdots & \vdots\\
a_{m1} & a_{m2} & \ldots & a_{mn} & b_m\\
\end{array}%
\right]
\]
is called the \emph{augmented matrix}. Some times we also denote augmented matrix by $C$.

Let us denote rank of coefficient matrix by $r(A)$ and  rank of augmented matrix by $r(C)$  then we have
\begin{enumerate}
\item If $r(A) \neq r(C)$, the system of equations is inconsistent.
\item If $r(A) = r(C)$ , the system of equations is consistent.
    \begin{enumerate}
    \item If rank is $n$, system possess a unique solution.
    \item If rank is less than $n$, system has infinite number of solutions.
    \end{enumerate}
\end{enumerate}
\section{Homogeneous linear equation system}
Let the system of $m$ linear  equations in $n$ unknowns is:
\begin{equation} \label{homoeq}
\begin{array}{cc}
a_{11}x_1 + a_{12}x_2 + \ldots + a_{1n}x_n &=0 \\
a_{21}x_1 + a_{22}x_2 + \ldots + a_{2n}x_n &=0 \\
\vdots &  \\
a_{m1}x_1 + a_{m2}x_2 + \ldots + a_{mn}x_n &=0 \\
\end{array}
\end{equation}
The system \ref{homoeq} may be written as:
\[ AX = O\]
Where
\[
A = \left[%
\begin{array}{rrrr}
a_{11} & a_{12} & \ldots & a_{1n} \\
a_{21} & a_{22} & \ldots & a_{2n} \\
\vdots & \vdots & \vdots & \vdots \\
a_{m1} & a_{m2} & \ldots & a_{mn} \\
\end{array}%
\right]
\]
and augmented matrix
\[
C = \left[%
\begin{array}{ccccc}
a_{11} & a_{12} & \ldots & a_{1n} & 0\\
a_{21} & a_{22} & \ldots & a_{2n} & 0\\
\vdots & \vdots & \vdots & \vdots &  \\
a_{m1} & a_{m2} & \ldots & a_{mn} & 0\\
\end{array}%
\right]
\]
It is obvious that rank of $A$ and the rank of $r(C)$ is always same. Hence the system of
equations is always consistent.
    \begin{enumerate}
    \item If rank is $n$, system possess a unique solution that is trivial solution given as $x_1=x_2= ... = x_n =0$.
    \item If rank is less than $n$, system has infinite number of solutions (These are non-trivial).
    \end{enumerate}
    
%*********************************************************************************
\section*{Problems}
\begin{enumerate}
\item  Solve the following equations: \\ $x-y + 2z = 3$, $x + 2y  + 3z = 5$, ${3x-4y-5z=-13}$ 

\item   Find the solutions of the equations: $x_1 + 2x_2 - x_3 = 1$, $3x_1 - 2x_2 + 2x_3 = 2$, $7x_1-2x_2 + 3x_3 = 5$. 

\item   Express the following system of equations in matrix form and solve them by the elimination method (Gauss Jordan Method) $2x_1 + x_2 + 2x_3 + x_4 = 6$, $6x_1-6x_2 + 6x_3 + 12x_4 = 36$, $4x_1 + 3x_2 + 3x_3 - 3x_4 = -1$, $2x_1 + 2x_2 - x_3 + x_4 = 10$.

\item   Find the general solution of the system of equations $3x_1 + 2x_3 + 2x_4 = 0$, $-x_1 + 7x_2 + 4x_3 + 9x_4 = 0$, $7x_1 -7x_2 - 5x_4 = 0$. 

\item   Show that the equations $2x + 6y = -11$, $6x + 20y - 6z = -3$, $6y - 18z = -1$ are not consistent.

\item   Test for consistency and solve $5x + 3y + 7z = 4$, $3x + 26y + 2z = 9$, $7x + 2y + 10z = 5$.

\item   Test the consistency of the following system of linear equations and hence find the solution. $4x_1 - x_2 = 12$, $-x_1 + 5x_2 - 2x_3 = 0$, $-2x_2 + 4x_3 = -8$.

\item   Using matrix method, show that the equations $3x + 3y + 2z = 1$, $x + 2y = 4$, $10y + 3z = -2$, $2x - 3y - z = 5$ are consistent and hence obtain the solution for $x, y$ and $z$. 

\item   Test for consistency the following system of equations and, if consistent, Solve them. $x_1 + 2x_2 - x_3 = 3$, $3x_1 - x_2 + 2x_3 = 1$, $2x_1 - 2x_2 + 3x_3 = 2$, $x_1 - x_2 + x_3 = -1$.

\item   Discuss the consistency of the following system of equations $2x + 3y + 4z = 11$, $x + 5y + 7z = 15$, $3x + 11y + 13z = 25$. If found consistent, solve it.

\item   Determine for what values of $\lambda $ and $\mu $ the following equations have (i) no solution (ii) a unique solution (iii) infinite number of solution. $x + y + z = 6$, $x + 2y + 3z = 10$, $x + 2y + \lambda z = \mu $. 

\item   Find for what values of $k$ the set of equations $2x - 3y + 6z - 5t = 3$, $y - 4z + t = 1$, $4x - 5y + 8z - 9t = k$ has (i) no solution (ii) infinite number of solutions.

\item   Determine the value of $\lambda$ so that the equations $2x + y + 2z = 0$, $x + y + 3z = 0$, $4x + 3y + \lambda z = 0$ have a non-zero solution. 

\item   Find the values of $k$ such that the system of equations $x + ky + 3z = 0$, $4x + 3y + kz = 0$, $2x + y + 2z = 0$ has non-trivial solution. 

\item   Find values of $\lambda$ for which the equations: $(\lambda -1) x + (3\lambda +1) y + 2\lambda z = 0$, $4(\lambda -1) x + (4\lambda -2) y +(\lambda +3) z = 0$, $2x + (3\lambda +1) y + 3(\lambda -1)z = 0$ are consistent, and find the ratio of $x : y : z$ when $\lambda$ has the smallest of these values. What happens when $\lambda$ has the greatest of these values.

\item   Find the values of $\lambda$ such that following equations have unique solution:  $\lambda x + 2y - 2z - 1 = 0$, $4x + 2\lambda y -- z - 2 = 0$, $6x + 6y +\lambda z - 3 = 0$. 

\item   Test the consistency and hence solve the following set of equations. $x_1 + 2x_2 + x_3 = 2$, $3x_1 + x_2 - 2x_3 =1$, $4x_1 - 3x_2 - x_3 = 3$, $2x_1 + 4x_2 + 2x_3 = 4$. 
\end{enumerate}

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